Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3

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Theorem

Let $\sequence {x_n}$ be the real sequence defined as $x_n = \paren {n + 1}^{1/n}$, using exponentiation.

Then $\sequence {x_n}$ converges with a limit of $1$.


Proof

We have the definition of the power to a real number:

$\paren {n + 1}^{1/n} = \map \exp {\dfrac 1 n \, \map \ln {n + 1} }$


For $n >= 1$ then $n + 1 \le 2 n$.

Hence:

\(\displaystyle \frac 1 n \, \map \ln {n + 1}\) \(\le\) \(\displaystyle \frac 1 n \, \map \ln {2 n}\) Logarithm is Strictly Increasing
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 n \paren {\ln 2 + \ln n}\) Logarithm on Positive Real Numbers is Group Isomorphism
\(\displaystyle \) \(\) \(\displaystyle \)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\ln 2} n + \frac 1 n \ln n\)


By Powers Drown Logarithms:

$\displaystyle \lim_{n \mathop \to \infty} \frac 1 n \ln n = 0$


By Sequence of Reciprocals is Null Sequence:

$\displaystyle \lim_{n \mathop \to \infty} \frac 1 n = 0$


By Combined Sum Rule for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \paren {\frac {\ln 2} n + \frac 1 n \ln n} = \ln 2 \cdot 0 + 0 = 0$


By the Squeeze Theorem for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} = 0$


Hence:

$\displaystyle \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} = \exp 0 = 1$

and the result follows.

$\blacksquare$